Is God a Mathematician?

by Mario Livio

  Another major contribution attributed to the Pythagoreans was the sobering discovery that their own “numerical religion” was, in fact, pitifully unworkable. The whole numbers 1, 2, 3,…are insufficient even for the construction of mathematics, let alone for a description of the universe. Examine the square in figure 6, in which the length of the side is one unit, and where we denote the length of the diagonal by d. We can easily find the length of the diagonal, using the Pythagorean theorem in any of the two right triangles into which the square is divided. According to the theorem, the square of the diagonal (the hypotenuse) is equal to the sum of the squares of the two shorter sides: d2 = 12 + 12, or d2 = 2. Once you know the square of a positive number, you find the number itself by taking the square root (e.g., if x2 = 9, then the positive x = v 9 = 3). Therefore, d2 = 2 implies d = v2 units. So the ratio of the length of the diagonal to the length of the square’s side is the number v2. Here, however, came the real shock—a discovery that demolished the meticulously constructed Pythagorean discrete-number philosophy. One of the Pythagoreans (possibly Hippasus of Metapontum, who lived in the first half of the fifth century BC) managed to prove that the square root of two cannot be expressed as a ratio of any two whole numbers. In other words, even though we have an infinity of whole numbers to choose from, the search for two of them that give a ratio of v2 is doomed from the start. Numbers that can be expressed as a ratio of two whole numbers (e.g., 3/17; 2/5; 1/10; 6/1) are called rational numbers. The Pythagoreans proved that v2 is not a rational number. In fact, soon after the original discovery it was realized that neither are v3, v17, or the square root of any number that is not a perfect square (such as 16 or 25). The consequences were dramatic—the Pythagoreans showed that to the infinity of rational numbers we are forced to add an infinity of new kinds of numbers—ones that today we call irrational numbers. The importance of this discovery for the subsequent development of mathematical analysis cannot be overemphasized. Among other things, it led to the recognition of the existence of “countable” and “uncountable” infinities in the nineteenth century. The Pythagoreans, however, were so overwhelmed by this philosophical crisis that the philosopher Iamblichus reports that the man who discovered irrational numbers and disclosed their nature to “those unworthy to share in the theory” was “so hated that not only was he banned from [the Pythagoreans’] common association and way of life, but even his tomb was built, as if [their] former comrade was departed from life among mankind.”

  Perhaps even more important than the discovery of irrational numbers was the pioneering Pythagorean insistence on mathematical proof—a procedure based entirely on logical reasoning, by which starting from some postulates, the validity of any mathematical proposition could be unambiguously established. Prior to the Greeks, even mathematicians did not expect anyone to be interested in the least in the mental struggles that had led them to a particular discovery. If a mathematical recipe worked in practice—say for divvying up parcels of land—that was proof enough. The Greeks, on the other hand, wanted to explain why it worked. While the notion of proof may have first been introduced by the philosopher Thales of Miletus (ca. 625–547 BC), the Pythagoreans were the ones who turned this practice into an impeccable tool for ascertaining mathematical truths. The significance of this breakthrough in logic was enormous. Proofs stemming from postulates immediately put mathematics on a much firmer foundation than that of any other discipline discussed by the philosophers of the time. Once a rigorous proof, based on steps in reasoning that left no loopholes, had been presented, the validity of the associated mathematical statement was essentially unassailable. Even Arthur Conan Doyle, the creator of the world’s most famous detective, recognized the special status of mathematical proof. In A Study in Scarlet, Sherlock Holmes declares that his conclusions are “as infallible as so many propositions of Euclid.”

  On the question of whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt—mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the feeble human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician—mathematics was God!

  The importance of the Pythagorean philosophy lies not only in its actual, intrinsic value. By setting the stage, and to some extent the agenda, for the next generation of philosophers—Plato in particular—the Pythagoreans established a commanding position in Western thought.

  Into Plato’s Cave

  The famous British mathematician and philosopher Alfred North Whitehead (1861–1947) remarked once that “the safest generalization that can be made about the history of western philosophy is that it is all a series of footnotes to Plato.”

  Indeed, Plato (ca. 428–347 BC) was the first to have brought together topics ranging from mathematics, science, and language to religion, ethics, and art and to have treated them in a unified manner that essentially defined philosophy as a discipline. To Plato, philosophy was not some abstract subject, divorced from everyday activities, but rather the chief guide to how humans should live their lives, recognize truths, and conduct their politics. In particular, he maintained that philosophy can gain us access into a realm of truths that lies far beyond what we can either perceive directly with our senses or even deduce by simple common sense. Who was this relentless seeker of pure knowledge, absolute good, and eternal truths?

  Plato, the son of Ariston and Perictione, was born in Athens or Aegina. Figure 7 shows a Roman herm of Plato that was most likely copied from an older, fourth century BC Greek original. His family had a long line of distinction on both sides, including such figures as Solon, the celebrated lawmaker, and Codrus, the last king of Athens. Plato’s uncle Charmides and his mother’s cousin Critias were old friends of the famous philosopher Socrates (ca. 470–399 BC)—a relation that in many ways defined the formative influence to which the young Plato’s mind was exposed. Originally, Plato intended to enter into politics, but a series of violent actions by the political faction that courted him at the time convinced him otherwise. Later in life, this initial repulsion by politics may have encouraged Plato to outline what he regarded as the essential education for future guardians of the state. In one case, he even attempted (unsuccessfully) to tutor the ruler of Syracuse, Dionysius II.

  Figure 7

  Following the execution of Socrates in 399 BC, Plato embarked on extensive travel that ended only when he founded his renowned school of philosophy and science—the Academy—around 387 BC. Plato was the director (or scholarch) of the Academy until his death, and his nephew Speusippus succeeded him in that position. Unlike academic institutions today, the Academy was a rather informal gathering of intellectuals who, under Plato’s guidance, pursued a wide variety of interests. There were no tuition fees, no prescribed curricula, and not even real faculty members. Still, there was apparently one rather unusual “entrance requirement.” According to an oration by the fourth century (AD) emperor Julian the Apostate, a burdensome inscription hung over the door to Plato’s Academy. While the text of the inscription does not appear in the oration, it can be found in another fourth century marginal note. The inscription read: “Let no one destitute of geometry enter.” Since no fewer than eight centuries separate the establishment of the Academy and the first description of the inscription, we cannot be absolutely certain that such an inscription indeed existed. There is no doubt, however, that the sentiment expressed by this demanding requirement reflected Plato’s personal opinion. In one of his famous dialogues, Gorgias, Plato writes: “Geometric equality is of great importance among gods and men.”

  The “students” in the Academy were generally self-supporting, and some of them—the great Aristotle for one—stayed there for as long as twenty years. Plato considered this long-term contact of creative minds to be the best vehicle for the production of new ideas, in topics ranging from abstract metaphysics and mathematics to ethics and politics. The purity and almost
divine attributes of Plato’s disciples were captured beautifully in a painting entitled The School of Plato by the Belgian symbolist painter Jean Delville (1867–1953). To emphasize the spiritual qualities of the students, Delville painted them in the nude, and they appear to be androgynous, because that was supposed to be the state of primordial humans.

  I was disappointed to discover that archaeologists were never able to find the remains of Plato’s Academy. On a trip to Greece in the summer of 2007, I looked for the next best thing. Plato mentions the Stoa of Zeus (a covered walkway built in the fifth century BC) as a favorite place to talk to friends. I found the ruins of this stoa in the northwest part of the ancient agora in Athens (which was the civic center in Plato’s time; figure 8). I must say that even though the temperature reached 115 °F that day, I felt something like a shiver as I walked along the same path that must have been traversed hundreds, if not thousands of times by the great man.

  Figure 8

  The legendary inscription above the Academy’s door speaks loudly about Plato’s attitude toward mathematics. In fact, most of the significant mathematical research of the fourth century BC was carried out by people associated in one way or another with the Academy. Yet Plato himself was not a mathematician of great technical dexterity, and his direct contributions to mathematical knowledge were probably minimal. Rather, he was an enthusiastic spectator, a motivating source of challenge, an intelligent critic, and an inspiring guide. The first century philosopher and historian Philodemus paints a clear picture: “At that time great progress was seen in mathematics, with Plato serving as the general architect setting out problems, and the mathematicians investigating them earnestly.” To which the Neoplatonic philosopher and mathematician Proclus adds: “Plato…greatly advanced mathematics in general and geometry in particular because of his zeal for these studies. It is well known that his writings are thickly sprinkled with mathematical terms and that he everywhere tries to arouse admiration for mathematics among students of philosophy.” In other words, Plato, whose mathematical knowledge was broadly up to date, could converse with the mathematicians as an equal and as a problem presenter, even though his personal mathematical achievements were not significant.

  Another striking demonstration of Plato’s appreciation of mathematics comes in what is perhaps his most accomplished book, The Republic, a mind-boggling fusion of aesthetics, ethics, metaphysics, and politics. There, in book VII, Plato (through the central figure of Socrates) outlined an ambitious plan of education designed to create utopian state rulers. This rigorous if idealized curriculum envisaged an early training in childhood imparted through play, travel, and gymnastics. After the selection of those who showed promise, the program continued with no fewer than ten years of mathematics, five years of dialectic, and fifteen years of practical experience, which included holding commands in time of war and other offices “suitable to youth.” Plato gave clear explanations as to why he thought that this was the necessary training for the would-be politicians:

  What we require is that those who take office should not be lovers of rule. Otherwise there will be a contest with rival lovers. What others, then, will you compel to undertake the guardianship of the city than those who have most intelligence of the principles that are the means of good government and who possess distinctions of another kind and a life that is preferable to political life?

  Refreshing, isn’t it? In fact, while such a demanding program was probably impractical even in Plato’s time, George Washington agreed that an education in mathematics and philosophy was not a bad idea for the politicians-to-be:

  The science of figures, to a certain degree, is not only indispensably requisite in every walk of civilized life; but investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational being. In a clouded state of existence, where so many things appear precarious to the bewildered research, it is here that the rational faculties find foundation to rest upon. From the high ground of mathematical and philosophical demonstration, we are insensibly led to far nobler speculations and sublimer meditations.

  For the question of the nature of mathematics, even more important than Plato the mathematician or the math stimulator was Plato the philosopher of mathematics. There his trail-blazing ideas put him not only above all the mathematicians and philosophers of his generation, but identified him as an influential figure for the following millennia.

  Plato’s vision of what mathematics truly is makes strong reference to his famous Allegory of the Cave. There he emphasizes the doubtful validity of the information provided through the human senses. What we perceive as the real world, Plato says, is no more real than shadows projected onto the walls of a cavern. Here is the remarkable passage from The Republic:

  See human beings as though they were in an underground cave-like dwelling with an entrance, a long one, open to the light across the whole width of the cave. They are in it from childhood with their legs and necks in bonds so that they are fixed, seeing only in front of them, unable because of the bond to turn their heads all the way around. Their light is from a fire burning far above and behind them. Between the fire and the prisoners there is a road above, along which we see a wall, built like the partitions puppet-handlers set in front of the human beings and over which they show the puppets…Then also see along this wall human beings carrying all sorts of artifacts, which project above the wall, and statues of men and other animals wrought from stone, wood, and every kind of material…do you suppose such men would have seen anything of themselves and one another, other than the shadows cast by the fire on the side of the cave facing them?

  According to Plato, we, humans in general, are no different from those prisoners in the cave who mistake the shadows for reality. (Figure 9 shows an engraving by Jan Saenredam from 1604 illustrating the allegory.) In particular, Plato stresses, mathematical truths refer not to circles, triangles, and squares that can be drawn on a piece of papyrus, or marked with a stick in the sand, but to abstract objects that dwell in an ideal world that is the home of true forms and perfections. This Platonic world of mathematical forms is distinct from the physical world, and it is in this first world that mathematical propositions, such as the Pythagorean theorem, hold true. The right triangle we might draw on paper is but an imperfect copy—an approximation—of the true, abstract triangle.

  Another fundamental issue that Plato examined in some detail concerned the nature of mathematical proof as a process that is based on postulates and axioms. Axioms are basic assertions whose validity is assumed to be self-evident. For instance, the first axiom in Euclidean geometry is “Between any two points a straight line may be drawn.” In The Republic, Plato beautifully combines the concept of postulates with his notion of the world of mathematical forms:

  Figure 9

  I think you know that those who occupy themselves with geometries and calculations and the like, take for granted the odd and the even [numbers], figures, three kinds of angles, and other things cognate to these in each subject; assuming these things as known, they take them as hypotheses and thenceforward they do not feel called upon to give any explanation with regard to them either to themselves or anyone else, but treat them as manifest to every one; basing themselves on these hypotheses, they proceed at once to go through the rest of the argument till they arrive, with general assent, at the particular conclusion to which their inquiry was directed. Further you know that they make use of visible figures and argue about them, but in doing so they are not thinking about these figures but of the things which they represent; thus it is the absolute square and the absolute diameter which is the object of their argument, not the diameter which they draw…the object of the inquirer being to see their absolute counterparts which cannot be seen otherwise than by thought [emphasis added].

  Plato’s views formed the basis for what has become known in philosophy in general, and in discussions of the nature of mathematics in particular, as P
latonism. Platonism in its broadest sense espouses a belief in some abstract eternal and immutable realities that are entirely independent of the transient world perceived by our senses. According to Platonism, the real existence of mathematical objects is as much an objective fact as is the existence of the universe itself. Not only do the natural numbers, circles, and squares exist, but so do imaginary numbers, functions, fractals, non-Euclidean geometries, and infinite sets, as well as a variety of theorems about these entities. In short, every mathematical concept or “objectively true” statement (to be defined later) ever formulated or imagined, and an infinity of concepts and statements not yet discovered, are absolute entities, or universals, that can neither be created nor destroyed. They exist independently of our knowledge of them. Needless to say, these objects are not physical—they live in an autonomous world of timeless essences. Platonism views mathematicians as explorers of foreign lands; they can only discover mathematical truths, not invent them. In the same way that America was already there long before Columbus (or Leif Ericson) discovered it, mathematical theorems existed in the Platonic world before the Babylonians ever initiated mathematical studies. To Plato, the only things that truly and wholly exist are those abstract forms and ideas of mathematics, since only in mathematics, he maintained, could we gain absolutely certain and objective knowledge. Consequently, in Plato’s mind, mathematics becomes closely associated with the divine. In the dialogue Timaeus, the creator god uses mathematics to fashion the world, and in The Republic, knowledge of mathematics is taken to be a crucial step on the pathway to knowing the divine forms. Plato does not use mathematics for the formulation of some laws of nature that are testable by experiments. Rather, for him, the mathematical character of the world is simply a consequence of the fact that “God always geometrizes.”